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Euro. Jnl of Applied Mathematics (2002), vol. 13, pp. 545–566. c© 2002 Cambridge University Press

DOI: 10.1017/S095679250100465X Printed in the United Kingdom

545

Direct construction method for conservationlaws of partial differential equations

Part I: Examples of conservation lawclassifications

STEPHEN C. ANCO 1 and GEORGE BLUMAN 2

1 Department of Mathematics, Brock University, St. Catharines, ON Canada L2S 3A1

email: [email protected] Department of Mathematics, University of British Columbia, Vancouver, BC Canada V6T 1Z2

email: [email protected]

(Received 5 January 1999; revised 18 May 2001)

An effective algorithmic method is presented for finding the local conservation laws for

partial differential equations with any number of independent and dependent variables. The

method does not require the use or existence of a variational principle and reduces the

calculation of conservation laws to solving a system of linear determining equations similar

to that for finding symmetries. An explicit construction formula is derived which yields a

conservation law for each solution of the determining system. In the first of two papers (Part

I), examples of nonlinear wave equations are used to exhibit the method. Classification results

for conservation laws of these equations are obtained. In a second paper (Part II), a general

treatment of the method is given.

1 Introduction

In the study of differential equations, conservation laws have many significant uses,

particularly with regard to integrability and linearization, constants of motion, analysis of

solutions, and numerical solution methods. Consequently, an important problem is how

to calculate all of the conservation laws for given differential equations.

For a differential equation with a variational principle, Noether’s theorem [12, 4, 6, 5, 14]

gives a formula for obtaining the local conservation laws by use of symmetries of the

action. One usually attempts to find these symmetries by noting that any symmetry of the

action leaves invariant the extremals of the action and hence gives rise to a symmetry of

the differential equation. However, all symmetries of a differential equation do not necess-

arily arise from symmetries of the action when there is a variational principle. For example,

if a differential equation is scaling invariant, then the action is often not invariant. Indeed

it is often computationally awkward to determine the symmetries of the action and carry

out the calculation with the formula to obtain a conservation law. Moreover, in general

a differential equation need not have a variational principle even allowing for a change

of variables. Therefore, it is more effective to seek a direct, algorithmic method without

involving an action principle to find the conservation laws of a given differential equation.

546 S. C. Anco and G. Bluman

In earlier work [2], we presented an algorithmic approach replacing Noether’s theorem

so as to allow one to obtain all local conservation laws for any differential equation

whether or not it has a variational principle. Details of this approach for the situation of

Ordinary Differential Equations (ODEs) are given in Anco & Bluman Ref. [3]. Here we

concentrate on the situation of Partial Differential Equations (PDEs).

In the case of a PDE with a variational principle, the approach shows how to use the

symmetries of the PDE to directly construct the conservation laws. The symmetries of

a PDE satisfy a linear determining equation, for which there is a standard algorithmic

method [5, 14] to seek all solutions. There is also an invariance condition, involving just

the PDE and its symmetries, which is necessary and sufficient for a symmetry of a PDE

with a variational principle to correspond to a symmetry of the action. The invariance

condition can be checked by an algorithmic calculation and, in addition, leads to a direct

construction formula for a conservation law in terms of the symmetry and the PDE. This

approach makes no use of the variational principle for the PDE.

In the case of a PDE without a variational principle, the approach involves replacing

symmetries by adjoint symmetries of the PDE. The adjoint symmetries satisfy a linear

determining equation that is the adjoint of the determining equation for symmetries.

Geometrically, symmetries of a PDE describe motions on the solution space of the PDE.

Adjoint symmetries in general do not have such an interpretation. The invariance con-

dition on symmetries is replaced by an adjoint invariance condition on adjoint symmetries

and there is a corresponding direct construction formula for obtaining the conservation

laws in terms of the adjoint symmetries and the PDE. The adjoint invariance condition

is a necessary and sufficient determining condition for an adjoint symmetry to yield a

conservation law.

In general for any PDE, with or without a variational principle, the approach of Anco

& Bluman Ref. [2] for finding all local conservation laws gave the following step-by-step

method:

(1) Find the adjoint symmetries of the given PDE.

(2) Check the adjoint invariance condition on the adjoint symmetries.

(3) For each adjoint symmetry satisfying the adjoint invariance condition use the direct

construction formula to obtain a conservation law.

If the adjoint symmetry determining equation is the same as the symmetry determining

equation then adjoint symmetries are symmetries. In this case the given PDE can be

shown to have a variational principle. Conversely, if a variational principle exists, then

the symmetry determining equation of the PDE can be shown to be self-adjoint, so that

symmetries are adjoint symmetries.

To solve the determining equation for adjoint symmetries, one works on the solution

space of the given PDE. On the other hand, in order to check the adjoint invariance

condition, one must move off the solution space by replacing the dependent variable(s)

of the given PDE by functions with arbitrary dependence on the independent variables

of the given PDE. The same situation arises when checking for invariance of the action

in Noether’s theorem.

These steps of the method are an algorithmic version of the standard treatment

presented in Olver Ref. [14] for finding PDE conservation laws in terms of multipliers. In

Direct conservation law method 547

particular, multipliers can be characterized as adjoint symmetries that satisfy the adjoint

invariance condition and thus can be calculated by the step-by-step algorithm (1), (2), (3).

In this paper (Part I) and a sequel (Part II), we significantly improve the effectiveness of

this method by replacing the adjoint invariance condition by extra determining equations

which allow one to work entirely on the solution space of the given PDE. Consequently,

by augmenting the adjoint symmetry determining equation by these extra determining

equations we obtain a linear determining system for finding only those adjoint symmetries

that are multipliers yielding conservation laws. At first sight it is natural to proceed as

in Anco & Bluman Ref. [2], by solving the adjoint symmetry determining equation and

then checking which of the solutions satisfy the extra determining equations. On the

other hand, all of the determining equations are on an equal footing, and hence there

is no requirement to solve the adjoint symmetry determining equation first. Indeed, as

illustrated later in the examples in this paper, it is much more effective to start with

the extra determining equations before considering the adjoint symmetry determining

equation. This is true even in the case when the given PDE has a variational principle.

In solving the determining system one works completely on the solution space of the

given PDE. Hence one can use the same algorithmic procedures as for solving symmetry

determining equations in order to solve the conservation law determining system. In

particular, existing symbolic manipulation programs [8] that calculate symmetries can

be readily adapted to calculate solutions of the conservation law determining system.

Moreover, for each solution one can directly obtain the resulting conservation law by

evaluating the construction formula working entirely on the solution space of the given

PDE.

The conservation law determining system together with the conservation law construc-

tion formula give a general, direct, computational method for finding the local conservation

laws of given PDEs. We refer to this as the direct conservation law method. As emphasized

above, its effectiveness stems from allowing the calculation of conservation laws to be

carried out algorithmically up to any given order by solving a linear determining system

without moving off the solution space of the PDEs. Compared to the standard treatment

of PDE conservation laws, the determining system solves a long-standing question of how

one can delineate necessary and sufficient determining equations to find multipliers by

working entirely on the solution space of the given PDE. Most important, by mingling

the adjoint symmetry equations with the extra equations in the determining system, one

can gain a significant computational advantage over the standard methods for finding

multipliers.

In the next section we illustrate the direct conservation law method through classifying

conservation laws for three PDE examples: a generalized Korteweg-de Vries equation, a

nonlinear wave-speed equation, and a class of nonlinear Klein-Gordon equations. The

classification results obtained are new in that they establish the completeness of certain

families of conservation laws which are of interest for these PDEs. These examples show

how to calculate all conservation laws up to a given order and also how to determine

which PDEs in a specified class admit conservation laws of a given type.

In the second paper (Part II), we present a general derivation of the conservation law

determining system and construction formula, and we also give a summary of the general

method.


2 Examples of conservation law classifications

Here we illustrate the use of our direct conservation law method on three PDE examples.

For each example we derive the conservation law determining system and use it to obtain

a classification result for conservation laws.

The first example is a generalized Korteweg-de Vries equation in physical form, for which

there is no direct variational principle. The ordinary Korteweg-de Vries (KdV) equation

is well-known to have local conservation laws of every even order [11], which can be

understood to arise from a recursion operator [13]. Using our conservation law determining

system, we derive a direct, complete classification for all conservation laws up to second

order for the generalized KdV equation. This example illustrates a general approach for

finding and classifying conservation laws for non-variational evolution equations.

The second example is a scalar wave equation with non-constant wave speed depending

on the wave amplitude. This wave equation has a variational principle and admits local

conservation laws for energy and momentum arising by Noether’s theorem from time-

and space-invariance of its corresponding action. By applying our conservation law deter-

mining system, we classify all wave speeds for which there are extra conservation laws of

first-order and obtain the resulting conserved quantities. This example illustrates a general

approach for classifying nonlinear evolution PDEs that admit extra conservation laws.

The third example is a general class of nonlinear Klein-Gordon equations. The class

includes the sine-Gordon equation, Liouville equation, and Tzetzeica equation, which

are known to be integrable equations [1] with local conservation laws up to arbitrarily

high orders, starting at first order. Through our conservation law method we give a

classification of nonlinear Klein-Gordon equations admitting at least one second-order

conservation law. As a by-product we obtain an integrability characterization for some of

the equations in this class. This example shows a general approach to classifying integrable

PDEs by means of conservation laws.

2.1 Generalized Korteweg-de Vries equations

Consider the generalized Korteweg-de Vries equation

G = ut + unux + uxxx = 0 (2.1)

with parameter n > 0. This is a first order evolution PDE which has no variational

principle directly in terms of u and which reduces for n = 1, 2 to the ordinary KdV

equation and modified KdV equation, respectively. Its symmetries with infinitesimal

generator Xu = η [5, 14] satisfy the determining equation

0 = Dtη + unDxη + nun−1uxη + D3xη when G = 0 (2.2)

where Dt = ∂t + ut∂u + utx∂ux + utt∂ut + · · · and Dx = ∂x + ux∂u + uxx∂ux + utx∂ut + · · · are

total derivative operators with respect to t and x. The adjoint of Eq. (2.2) is given by

0 = −Dtω − unDxω − D3xω when G = 0 (2.3)

which is the determining equation for the adjoint symmetries ω of the generalized KdV

equation. (Note, since these determining equations are not self-adjoint, the only solution

common to Eqs. (2.2) and (2.3) is ω = η = 0.)


The generalized KdV equation (2.1) itself has the form of a local conservation law

Dt(u) + Dx

(1

n+ 1un+1 + uxx

)= 0. (2.4)

We now consider, more generally, local conservation laws

DtΦt + DxΦ

x = 0 (2.5)

on all solutions u(t, x) of Eq. (2.1). Clearly, we are free without loss of generality to

eliminate any dependence on ut (and differential consequences) in the conserved densities

Φt, Φx. All nontrivial conserved densities in this form can be constructed from multipliers

Λ on the generalized KdV equation, analogous to integrating factors, where Λ depends

only on t, x, u, and x derivatives of u. In particular, by moving off the generalized KdV

solution space, we have

DtΦt + DxΦ

x = (ut + unux + uxxx)Λ0 + Dx(ut + unux + uxxx)Λ1 + · · · (2.6)

for some expressions Λ0, Λ1, . . . with no dependence on ut and differential consequences.

This yields (after integration by parts) the multiplier

DtΦt + Dx(Φ

x − Γ ) = (ut + unux + uxxx)Λ, Λ = Λ0 − DxΛ1 + · · · (2.7)

where Γ = 0 when u is restricted to be a generalized KdV solution. We now derive

an augmented adjoint symmetry determining system which completely characterizes all

multipliers Λ.

The definition for multipliers Λ(t, x, u, ∂xu, . . . , ∂pxu) is that (ut + unux + uxxx)Λ must be

a divergence expression for all functions u(t, x) (not just generalized KdV solutions). This

determining condition is expressed by

0 = Eu((ut + unux + uxxx)Λ)

= −DtΛ − unDxΛ − D3xΛ + (ut + unux + uxxx)Λu − Dx((ut + unux + uxxx)Λ∂xu )

+ · · ·+ (−1)pDpx((ut + unux + uxxx)Λ∂pxu) (2.8)

where Eu = ∂u − Dt∂ut − Dx∂ux + DtDx∂utx + D2x∂uxx + · · · is the standard Euler operator

which annihilates divergence expressions. Equation (2.8) is linear in ut, utx, utxx, . . . , and

thus the coefficients of ut and x derivatives of ut up to order p give rise to a split system

of determining equations for Λ. The system is found to consist of the adjoint symmetry

determining equation on Λ,

0 = −DtΛ − unDxΛ − D3xΛ (2.9)

and extra determining equations on Λ,

0 =

p∑k=1

(−Dx)kΛ∂kxu (2.10 a)

0 = (1− (−1)q)Λ∂qxu+

p∑k=q+1

k!

q!(k − q)!(−Dx)k−qΛ∂kxu , q = 1, . . . , p− 1 (2.10 b)

0 = (1− (−1)p)Λ∂pxu. (2.10 c)


Here Dt = ∂t− (uux +uxxx)∂u− (uux +uxxx)x∂ux + · · · is the total derivative operator which

expresses t derivatives of u through the generalized KdV equation (2.1). Consequently,

one is able to work on the space of generalized KdV solutions u(t, x) to solve the

determining system (2.9) and (2.10a–c) to find Λ(t, x, u, ∂xu, . . . , ∂pxu). The determining

system solutions are the multipliers that yield all nontrivial generalized KdV conservation

laws.

The explicit relation between multipliers Λ and conserved densities Φt, Φx for generalized

KdV conservation laws is summarized as follows. Given a conserved density Φt, we find

that a direct calculation of the ut terms in DtΦt + DxΦ

x yields the multiplier equation

DtΦt + DxΦ

x = (ut + unux + uxxx)Eu(Φt) + DxΓ (2.11)

where Eu = ∂u −Dx∂ux +D2x∂uxx + · · · is a restricted Euler operator and Γ is proportional

to ut + unux + uxxx and differential consequences. Thus we obtain the multiplier

Λ = Eu(Φt). (2.12)

Conversely, given a multiplier Λ, we can invert the relation (2.12) by a standard method

[14] using Eq. (2.11) to obtain the conserved density

Φt =

∫ 1

0

dλ uΛ(t, x, λu, λ∂xu, λ∂2xu, . . .). (2.13)

From Eqs. (2.12) and (2.13) it is natural to define the order of a generalized KdV

conservation law as the order of the highest x derivative of u in its multiplier. Thus, we

see that all nontrivial generalized KdV conservation laws up to order p are determined by

multipliers of order p which are obtained as solutions of the augmented system of adjoint

symmetry determining equations (2.9) and (2.10a–c).

Through the determining system (2.9) and (2.10a–c) we now derive a complete classifi-

cation of all conservation laws (2.5) up to second order, corresponding to multipliers of

the form

Λ(t, x, u, ux, uxx). (2.14)

The classification results are summarized by the following theorem.

Theorem 2.1.1 The generalized KdV equation (2.1) for all n > 0 admits the multipliers

Λ = 1, Λ = u, Λ = uxx +1

n+ 1un+1. (2.15)

The only additional admitted multipliers of the form (2.14) are given by

Λ = tu− x, if n = 1, (2.16)

Λ = t

(uxx +

1

3u3

)− 1

3xu, if n = 2. (2.17)

This classifies all nontrivial conservation laws up to second order for the generalized KdV

equation for any n > 0.

The conserved densities for these conservation laws are easily obtained using the


construction formula (2.13) as follows. For the multipliers (2.15) we find, respectively,

Φt = u, (2.18)

Φt =1

2u2, (2.19)

Φt = −1

2ux

2 +1

(n+ 1)(n+ 2)un+2 + Dxθ, (2.20)

where Dxθ is a trivial conserved density. In physical terms, if we regard u as a wave

amplitude as in the ordinary (n = 1) KdV equation, then these conserved densities

represent mass, momentum and energy [15].

For the additional multipliers (2.16) and (2.17) we find

Φt =1

2tu2 − xu, if n = 1, (2.21)

Φt = −1

2tux

2 +1

12tu4 − 1

6xu2 + Dxθ, if n = 2. (2.22)

Proof of Theorem 2.1.1 For multipliers of the form (2.14) the adjoint symmetry equation

(2.9) becomes

0 = −D3xΛ− unDxΛ− Λt + Λu(u

nux + uxxx) + Λux (unuxx + nun−1ux

2 + uxxxx)

+Λuxx (unuxxx + 3nun−1uxuxx + n(n− 1)un−2ux

3 + uxxxxx) (2.23)

and the extra equations (2.10a–c) reduce to

0 = −DxΛux + D2xΛuxx , (2.24)

0 = Λux − DxΛuxx . (2.25)

Note that Eq. (2.24) is a differential consequence of Eq. (2.25).

We start from equation (2.25). Its term with highest order derivatives is uxxxΛuxxuxx , and

hence Λuxxuxx = 0. This yields that Λ is linear in uxx,

Λ = a(t, x, u, ux)uxx + b(t, x, u, ux). (2.26)

Then the remaining terms in Eq. (2.25), after some cancellations, are of first order

0 = bux − auux − ax. (2.27)

We now turn to the adjoint symmetry equation (2.23) and separate it into highest

derivative terms in descending order. The highest order terms cancel. The second highest

order terms involve uxxxx, and these yield

0 = Dxa = ax + auux + auxuxx. (2.28)

Clearly, Eq. (2.28) separates into

aux = au = ax = 0. (2.29)

Hence bux = 0 from Eq. (2.27).

The next highest order terms remaining in the adjoint symmetry equation (2.23) involve

uxx. These terms yield

0 = −at − 3bxu + 3(nun−1a− buu)ux (2.30)


which separates into buu = nun−1a and at = −3bxu. Hence we have

b =1

n+ 1aun+1 +

(b1(t)− 1

3xat

)u+ b2(t, x). (2.31)

This simplifies the remaining terms in Eq. (2.23),

0 = −b2xxx − b2t −(b1t − 1

3xatt

)u+ b2xu

n +

(1

3− 1

n+ 1

)atu

n+1. (2.32)

Clearly, the terms here can be separated, yielding

(n− 2)at = 0, (2.33)

b2t = −b2xxx, (2.34)

b1t − 1

3xatt = −b2xu

n−1. (2.35)

From Eq. (2.35) there are two cases to consider.

Case (i): n = 1

Here Eq. (2.33) yields

at = 0 (2.36)

and hence Eq. (2.35) becomes b1t = −b2x. Since b1 depends only on t, using Eq. (2.34) we

find b2xx = b2t = 0, and thus

b1 = b3t+ b4, b2 = −b3x+ b5, b3 = const, b4 = const, b5 = const. (2.37)

Finally, from Eqs. (2.29) and (2.36) it follows that

a = const. (2.38)

Therefore, through Eq. (2.26), Eq. (2.31), Eqs. (2.37) and (2.38), we have

Λ =1

2au2 + auxx + b3 − b5x+ (b4 + b5t)u, if n = 1, (2.39)

which is a linear combination of the multipliers (2.15) and (2.16) shown in Theorem 2.1.1.

Case (ii): n > 1

In this case Eq. (2.35) clearly splits into

b1t = att = 0, (2.40)

b2x = 0. (2.41)

Hence from Eq. (2.41) and Eq. (2.34) we have

b2 = const, (2.42)

and from Eq. (2.40) and Eq. (2.29) we have

a = a1t+ a2, a1 = const, a2 = const, (2.43)

b1 = const. (2.44)


Therefore, Eq. (2.31) becomes

b =1

n+ 1(a1t+ a2)un+1 +

(b1 − 1

3xa1

)u+ b2. (2.45)

Finally, consider Eq. (2.33). If n� 2 then we have at = 0 and hence a1 = 0. Otherwise,

if n = 2 then we have no restriction on a1. Consequently, through Eq. (2.26), Eqs. (2.43)

and (2.45), we find

Λ = (a1t+ a2)uxx +1

n+ 1un+1(a1t+ a2) +

(b1 − 1

3xa1

)u+ b2, if n > 1, (2.46)

where a1 = 0 for all n > 2. This yields a linear combination of the multipliers (2.15)

and (2.17) shown in Theorem 2.1.1. q

2.2 Nonlinear wave-speed equation

Consider the wave equation

G = utt − c(u)(c(u)ux)x = 0 (2.47)

with a non-constant wave speed c(u). This is a scalar second-order evolution PDE, which

has a variational principle given by the physically-motivated action

S =

∫1

2(−ut2 + c(u)2ux

2)dtdx. (2.48)

Symmetries of the wave equation (2.47) with infinitesimal generator Xu = η [5, 14] satisfy

the determining equation

0 = D2t η−c(u)2D2

xη−2c(u)c′(u)uxDxη−2c(u)c′(u)uxxη−(c(u)c′′(u)+c′(u)2)ux2η when G = 0.

(2.49)

Since Eq. (2.47) is variational, the determining equation (2.49) is self-adjoint and hence

the adjoint symmetries of the wave equation are the solutions of Eq. (2.49).

For any wave speed c(u), the wave equation (2.47) clearly admits time and space

translation symmetries

η = ut, η = ux, (2.50)

which are symmetries of the action. In particular, the action is invariant up to a boundary-

term

XS =

∫1

2Dt(−ut2 + c(u)2ux

2)dtdx (2.51)

under time-translation Xu = ut, and

XS =

∫1

2Dx(−ut2 + c(u)2ux

2)dtdx (2.52)

under space-translation Xu = ux. By Noether’s theorem, combining the invariance of the

action with the general variational identity

XS =

∫ ((utt − c2uxx − cc′ux2)η + Dt(−utη) + Dx(c

2uxη))dtdx, Xu = η, (2.53)

we obtain corresponding local conservation laws DtΦt +DxΦ

x = 0 on all solutions u(t, x)


of the wave equation (2.47). The conserved densities Φt, Φx are given by

Φt =1

2ut

2 +1

2c(u)2ux

2, Φx = −c(u)2uxut (2.54)

for the time-translation, and

Φt = uxut, Φx = −1

2ut

2 − 1

2c(u)2ux

2 (2.55)

for the space-translation. These yield conservation of energy and momentum, respectively.

We now consider classifying wave speeds c(u) that lead to additional local conservation

laws of first-order for the wave equation (2.47). (Through Noether’s theorem all first-order

conservation laws correspond to invariance of the action S under contact symmetries.)

For any local conservation laws

DtΦt + DxΦ

x = 0 (2.56)

on all solutions u(t, x) of the wave equation (2.47), we are clearly free without loss of

generality to eliminate any dependence on utt (and differential consequences) in the con-

served densities Φt, Φx. All nontrivial conserved densities in this form can be constructed

from multipliers Λ on the wave equation, analogous to integrating factors, where Λ has

no dependence on utt and its differential consequences. In particular, by moving off the

wave equation solution space, we have

DtΦt + DxΦ

x = (utt − c(u)(c(u)ux)x)Λ0 + Dx(utt − c(u)(c(u)ux)x)Λ1 + · · · (2.57)

for some expressions Λ0, Λ1, . . . with no dependence on utt and differential consequences.


DtΦt + Dx(Φ

x − Γ ) = (utt − c(u)(c(u)ux)x)Λ, Λ = Λ0 − DxΛ1 + · · · (2.58)

where Γ = 0 when u is restricted to be a wave equation solution. We now derive an

augmented symmetry determining system which completely characterizes all multipliers Λ.

Multipliers Λ are defined by the condition that (utt − c(u)(c(u)ux)x)Λ is a divergence

expression for all functions u(t, x) (not just wave equation solutions). We restrict attention

to Λ of first-order, depending on t, x, u, ut, ux. This leads to the necessary and sufficient

determining condition

0 = Eu((utt − c(u)(c(u)ux)x)Λ)

= D2t Λ − c2D2

xΛ − 2cc′uxDxΛ − 2cc′uxxΛ − (cc′′ + c′2)ux2Λ

+(utt − c2uxx − cc′ux2)Λu − Dt((utt − c2uxx − cc′ux2)Λut )

−Dx((utt − c2uxx − cc′ux2)Λux ) (2.59)

where Eu = ∂u − Dt∂ut − Dx∂ux + DtDx∂utx + D2x∂uxx + · · · is the standard Euler operator

which annihilates divergence expressions. Equation (2.59) is quadratic in utt and linear

in uttt and uttx, and thus splits into separate equations. We organize the splitting by

considering terms in G2, G, DxG, DtG, and remaining terms with no dependence on utt

and differential consequences. The coefficients of DxG, DtG, and G2 are found to vanish

as a result of Λ(t, x, u, ut, ux) being first-order. The other coefficients in the splitting do not

vanish.


This leads to a split system of two determining equations for Λ(t, x, u, ut, ux), consisting

of

0 = D2t Λ− c(u)2D2

xΛ− 2c(u)c′(u)uxDxΛ− 2c(u)c′(u)uxxΛ− (c(u)c′′(u) + c′(u)2)ux2Λ (2.60)

which is the symmetry determining equation (2.49) on Λ, and

0 = 2Λu +DtΛut − DxΛux (2.61)

which is an extra determining equation on Λ. Here Dt = ∂t + ut∂u + utx∂ux + (c(u)2uxx +

c(u)c′(u)ux2)∂ut + · · · is the total derivative operator which expresses t derivatives through

the wave equation (2.47). Consequently, one is able to work on the space of wave

equation solutions u(t, x) in order to solve the determining system (2.60) and (2.61) to

find Λ(t, x, u, ut, ux). The determining system solutions are the multipliers that yield all

nontrivial first-order conservation laws of the wave equation (2.47).

The first determining equation (2.60) shows that all multipliers are symmetries of the

wave equation, while the second determining equation (2.61) provides the necessary and

sufficient condition for a symmetry to leave the action (2.48) invariant up to a boundary

term. This is a consequence of the one-to-one correspondence between symmetries of the

action and multipliers for nontrivial conservation laws of the wave equation (2.47) as

shown by Noether’s theorem [14].

The explicit relation between multipliers Λ and conserved densities Φt, Φx for first-order

conservation laws (2.56) is summarized as follows. Given a conserved density Φt, we find

that a direct calculation of the utt terms in DtΦt + DxΦ


DtΦt + DxΦ

x = (utt − c(u)(c(u)ux)x)Eut(Φt) + DxΓ (2.62)

where Eut = ∂ut is the truncation of a restricted Euler operator, and Γ is proportional to

utt − c(u)(c(u)ux)x and differential consequences. Thus we obtain the multiplier

Λ = Eut(Φt). (2.63)

Conversely, given a multiplier Λ, we can invert the relation (2.63) using Eq. (2.62) to

obtain the conserved density

Φt =

∫ 1

0

dλ(

(ut− ut)Λ[λu+(1−λ)u]+(u−u)DtΛ[λu+(1−λ)u])

+t

∫ 1

0

dλK(λt, λx) (2.64)

where

Λ[u] = Λ(t, x, u, ut, ux), (2.65)

DtΛ[u] = Λt(t, x, u, ut, ux) + utΛu(t, x, u, ut, ux) + utxΛux (t, x, u, ut, ux)

+c(u)(c(u)ux)xΛut (t, x, u, ut, ux), (2.66)

K(t, x) = (utt − c(u)(c(u)ux)x)Λ[u], (2.67)

with u = u(t, x) being any function chosen such that the expressions Λ[u] and utt −c(u)(c(u)ux)x are non-singular.

Thus, we see that all nontrivial first-order conservation laws of the wave equation

(2.47) are determined by multipliers of first-order which are obtained as solutions of the

augmented system of symmetry determining equations (2.60) and (2.61).


We now use the determining system (2.60) and (2.61) to completely classify all first-order

conservation laws in terms of corresponding multipliers

Λ(t, x, u, ut, ux). (2.68)

The classification results are summarized by the following two theorems.

Theorem 2.2.1 For arbitrary wave speeds c(u), the only multipliers of form (2.68) admitted

by the wave equation (2.47) are

Λ = ut, Λ = ux, Λ = tut + xux. (2.69)

These multipliers define symmetries given by time-translation, space-translation, and

time-space dilation, respectively, which lead to conservation laws for energy (2.54), mo-

mentum (2.55), and a dilational quantity.

Theorem 2.2.2 The wave equation (2.47) for non-constant wave speed c(u) admits additional

multipliers of form (2.68) iff c(u) = c0(u− u0)−2 in terms of constants c0, u0. For these wave

speeds the additional admitted multipliers are given by

Λ = t2ut − t(u− u0), (2.70)

Λ = x2ux + x(u− u0), (2.71)

Λ = tut − xux − (u− u0). (2.72)

The symmetries defined by these additional multipliers correspond to two conformal

(Mobius) transformations and one scaling transformation on independent variables t, x,

accompanied by a scaling and shift of u. The conserved densities Φt for the three

corresponding conservation laws are obtained by the construction formula (2.64). We

choose u = const� u0 to avoid the singularity at u = u0. This leads to K = 0. Using

the linear dependence of Λ(t, x, u, ux, ut) together with the property Λxut = 0 given by

Eqs. (2.70) to (2.72), we find that after integration by parts the formula reduces to

Φt =1

2ut

(Λ[u] + Λ[u] + ((u − u)Λux [u])x

)+

1

2(u − u)

(Λt[u] + Λt[u] + utΛu[u]

)+

1

2ux

2c(u)2Λut [u] (2.73)

up to a trivial conserved density Dxθ. Since the terms in Eq. (2.73) are non-singular for

any constant u, we can now set u = u0 without loss of generality. Then, substituting

Eqs. (2.70) to (2.72) for Λ, we obtain, respectively,

Φt =1

2t2ut

2 − t(u− u0)ut +1

2(u− u0)2 +

1

2t2ux

2c02(u− u0)−4, (2.74)

Φt = x2uxut + x(u− u0)ut, (2.75)

Φt =1

2tut

2 − (xux + u− u0)ut +1

2tux

2c02(u− u0)−4, (2.76)

which represent two conformal quantities and a scaling quantity.


Proof of Theorems 2.2.1 and 2.2.2 We start from the determining equation (2.61) and

expand it in explicit form using

DtΛ = Λt + Λuut + Λuxutx + Λut (c2uxx + cc′ux2), (2.77)

DxΛ = Λx + Λuux + Λuxuxx + Λututx. (2.78)

This yields

0 = 2Λu + Λtut + Λuutut + cc′ux2Λutut − Λxux − Λuuxux + (c2Λutut − Λuxux )uxx. (2.79)

Since Λ does not depend on uxx, Eq. (2.79) separates into

0 = c2Λutut − Λuxux , (2.80)

0 = 2Λu + Λtut + Λuutut − Λxux − Λuuxux. (2.81)

Next we bring in the symmetry determining equation (2.60). Expanding it similarly, we

find after use of Eq. (2.80) that its highest derivative terms involve uxx and utx. Hence,

since Λ does not depend on second-derivatives of u, these terms can be separated, leading

to

0 = (Λtut + Λuutut − Λxux − Λuuxux)c+ (Λutut + Λuxux − Λ)c′ + Λututux2c2c′,

(2.82)

0 = Λtux + Λuuxut − (Λxut + Λuutux)c2 + Λuxutux

2cc′. (2.83)

To proceed, we combine Eqs. (2.81) and (2.82) to yield

0 = (Λutut + Λuxux − Λ)c′ − 2Λuc (2.84)

which is a first order linear PDE for Λ in u, ut, ux. The solution of Eq. (2.84) is given by

Λ = c−1/2f(t, x, α, β), α = c1/2ut, β = c1/2ux. (2.85)

Substituting Eq. (2.85) into Eq. (2.80), we obtain c2fαα = fββ which separates into

fαα = fββ = 0 (2.86)

since c(u)� const. Hence we have

f = a(t, x)α+ b(t, x)β + g(t, x)αβ + h(t, x). (2.87)

Next, we substitute Eqs. (2.87) and (2.85) into Eq. (2.83) to obtain

0 = bt − c2ax +1

2c−1/2c′(ut2 + c2ux

2)g. (2.88)

This immediately yields

g = 0, bt = ax = 0. (2.89)

Thus, so far we have

Λ = a(t)ut + b(x)ux + h(t, x)c(u)−1/2. (2.90)

Now, using Eq. (2.90) we see that Eqs. (2.81) and (2.82) both reduce to

0 = −c−3/2c′h+ at − bx. (2.91)


This leads to two cases to consider. If the wave speed c(u) is arbitrary, then we must have

the following Case (i):

at = bx, h = 0. (2.92)

Alternatively, the only other possibility is for the wave speed c(u) to satisfy a first-order

ODE, so we then have Case (ii):

c−3/2c′ = c = const, ch = at − bx. (2.93)

Note in this case we require c� 0 for c(u) to be non-constant.

Finally, we return to the symmetry determining equation (2.60) and consider the terms

that remain after we substitute Eq. (2.90). The analysis proceeds according to the two cases.

In Case (i), the remaining terms in Eq. (2.60) reduce to

0 = −c2bxxux + attut (2.94)

which separates into

bxx = att = 0. (2.95)

Solving this equation and using Eqs. (2.92) and (2.89), we obtain

a = a0 + a1t, b = b0 + a1x (2.96)

where a0, b0, a1 are constants. Consequently, from Eq. (2.90), we have

Λ = a0ut + b0ux + a1(tut + xux). (2.97)

Hence, these are the only multipliers of first-order admitted by the wave equation (2.47)

for arbitrary wave speeds c(u). This establishes Theorem 2.2.1.

Finally, in Case (ii), using Eq. (2.93) for h and Eq. (2.89) for a and b, then eliminating

c′ by Eq. (2.93), we find that the remaining terms in Eq. (2.60) simplify considerably to

0 = c−1/2htt − c−3/2hxx. (2.98)

Hence, since c(u) is non-constant, we obtain

htt = hxx = 0. (2.99)

Now we solve for a, b by combining Eq. (2.99) with Eqs. (2.93) and (2.89), which gives

attt = bxxx = 0. (2.100)

Hence we have

a = a0 + a1t+ a2t2, b = b0 + b1x+ b2x

2, (2.101)

where a0, a1, a2, b0, b1, b2 are constants. Then Eq. (2.93) yields

ch = a1 + 2a2t− b1 − 2b2x. (2.102)

Consequently, from Eq. (2.90), we obtain

Λ = a0ut + b0ux + a1(tut + c−1/2/c) + b1(xux − c−1/2/c)

+a2(t2ut + 2tc−1/2/c) + b2(x2ux − 2xc−1/2/c) (2.103)

where the wave speed is given from Eq. (2.93) by

c(u) = c0(u− u0)−2 (2.104)

with c = −2c0−1/2, c0 > 0.


In Eq. (2.103) we have a six parameter family of admitted multipliers. The param-

eters a0, b0, a1 = b1 yield the three multipliers of Case (i). Hence the present case has

three additional multipliers arising from the parameters a1 = −b1, a2, b2. This establishes

Theorem 2.2.2. q

2.3 Klein-Gordon wave equations

Consider the class of Klein-Gordon wave equations

G = utx − g(u) = 0 (2.105)

with a nonlinear interaction g(u). This class has a variational principle given by the action

S = −∫ (

1

2utux + h(u)

)dtdx, h′(u) = g(u). (2.106)

Since the general Klein-Gordon equation (2.105) is variational, its symmetries with in-

finitesimal generator Xu = η [5, 14] satisfy the determining equation

0 = DtDxη − g′(u)η (2.107)

which is self-adjoint. Hence the adjoint symmetries of Eq. (2.105) are the solutions of

Eq. (2.107).

The symmetries Xu = η admitted by the general Klein-Gordon equation (2.105) clearly

consist of t and x translations

η = ut, η = ux, (2.108)

as well as a t, x boost

η = tut − xux. (2.109)

These are easily checked to be symmetries of the action, leaving S invariant up to a

boundary-term, with

XS = −∫Dt

(1

2utux + h(u)

)dtdx, XS = −

∫Dx

(1

2utux + h(u)

)dtdx (2.110)

under the respective translations Xu = ut and Xu = ux, and with

XS =

∫ (−Dt

(1

2tutux + th(u)

)+ Dx

(1

2xutux + xh(u)

))dtdx (2.111)

under the boost Xu = tut − xux. By Noether’s theorem, combining the invariance of the

action and the general variational identity

XS =

∫ ((utx − g(u))η − Dt

(1

2ηux

)− Dx

(1

2ηut

))dtdx, Xu = η, (2.112)

we obtain corresponding local conservation laws DtΦt +DxΦ

x = 0 on all solutions u(t, x)

of the Klein-Gordon equation (2.105). The conserved densities are given by

Φt = h(u), Φx = −1

2ut

2, (2.113)

Φt = −1

2ux

2, Φx = h(u), (2.114)


from the translations, and

Φt =1

2xux

2 + th(u), Φx = −1

2tut

2 − xh(u), (2.115)

from the boost. These comprise all of the first-order local conservation laws for the general

Klein-Gordon equation (2.105).

The Klein-Gordon equations in the class (2.105) include

utx = sin u, sine-Gordon equation (2.116)

utx = eu ± e−u, sinh-Gordon equation (2.117)

utx = eu ± e−2u, Tzetzeica equation (2.118)

utx = eu. Liouville equation (2.119)

The first three are soliton equations while the last is a linearizable equation. These are

singled out [1] as nonlinear wave equations that are known to be integrable in the sense

of admitting an infinite number of higher-order local conservation laws [7, 10]. For each

equation the conservation laws fall into two sequences where Φt and Φx depend purely

on u and t derivatives of u in one sequence, and purely on u and x derivatives of u in the

other sequence (corresponding to the reflection symmetry t↔ x).

Here, for the class of nonlinear Klein-Gordon equations (2.105), we consider local

conservation laws of higher-order

DtΦt + DxΦ

x = 0 (2.120)

with Φt, Φx depending either purely on t, u, and t derivatives of u, or purely on x, u, and

x derivatives of u, on all solutions u(t, x) of Eq. (2.105).

All nontrivial conservation laws (2.120) with conserved densities of the particular form

Φt(x, u, ∂xu, . . . , ∂qxu), Φx(x, u, ∂xu, . . . , ∂

qxu) (2.121)

can be constructed from multipliers Λ on the Klein-Gordon equation (2.105), analogous

to integrating factors, with the dependence

Λ(x, u, ∂xu, . . . , ∂pxu) (2.122)

where p = 2q − 1. In particular, moving off the Klein-Gordon solution space, we have

DtΦt + DxΦ

x = (utx − g(u))Λ0 + Dx(utx − g(u))Λ1 + · · · (2.123)

for some expressions Λ0, Λ1, . . . with no dependence on ut and differential consequences.


DtΦt + Dx(Φ

x − Γ ) = (utx − g(u))Λ, Λ = Λ0 − DxΛ1 + · · · (2.124)

where Γ = 0 when u is restricted to be a Klein-Gordon solution. There are corresponding

conservation laws of mirror form produced by the transformation x↔ t, ∂kxu ↔ ∂kt u, with

Φt ↔ Φx. We now derive an augmented symmetry determining system which completely

characterizes the conservation law multipliers (2.122).

The determining condition for multipliers Λ is that (utx − g(u))Λ must be a divergence

expression for all functions u(t, x) (not just Klein-Gordon solutions). For multipliers of


the form (2.122), this condition is expressed by

0 = Eu((utx − g)Λ)

= DtDxΛ − g′Λ + Λu(utx − g)− Dx(Λ∂xu (utx − g)) + · · ·+ (−Dx)p(Λ∂pxu (utx − g))

(2.125)

where Eu = ∂u−Dt∂ut−Dx∂ux+DtDx∂utx+D2x∂uxx+D

2t ∂utt+· · · is the standard Euler operator

which annihilates divergence expressions. Equation (2.125) is linear in utx, utxx, . . . , and

hence splits into separate equations. We organize the splitting in terms of G,DxG, . . . , DpxG.

This yields a split system of determining equations for Λ(x, u, ∂xu, . . . , ∂pxu), which is found

to consist of the Klein-Gordon symmetry determining equation on Λ

0 = DtDxΛ − g′(u)Λ (2.126)

and extra determining equations on Λ

0 = 2Λu +

p∑k=2

(−Dx)kΛ∂kxu , (2.127)

0 = (1 + (−1)j)Λ∂jxu

+ ((−1)j − j − 1)DxΛ∂j+1x u

+

p∑k=j+2

k!

j!(k − j)! (−Dx)k−jΛ∂kxu , j = 1, . . . , p− 1 (2.128)

0 = (1 + (−1)p)Λ∂pxu. (2.129)

Here Dt = ∂t + ut∂u + g(u)∂ux + g′(u)ux∂uxx + · · · is the total derivative operator which

expresses t derivatives of u through the Klein-Gordon equation (2.105). Consequently,

one is able to work on the space of Klein-Gordon solutions u(t, x) in order to solve the

determining system (2.126) to (2.129) to find Λ(x, u, ∂xu, . . . , ∂pxu).

The solutions of the determining system (2.126) to (2.129) are the multipliers (2.122) that

yield all nontrivial conservation laws of the form (2.121). Noether’s theorem establishes

that there is a one-to-one correspondence between symmetries of the action and multipliers

for nontrivial conservation laws of the Klein-Gordon equation (2.105). Consequently, it

follows that the extra determining equations (2.127)–(2.129) represent the necessary and

sufficient condition for a symmetry to leave the action (2.106) invariant up to a boundary

term.

The explicit relation between multipliers Λ and conserved densities Φt, Φx for conser-

vation laws (2.121) is summarized as follows. Given a conserved density Φt, we find that

a direct calculation of the utx terms in DtΦt + DxΦ


DtΦt + DxΦ

x = (utx − g(u))Eux(Φt) + DxΓ (2.130)

where Eux = ∂ux − Dx∂uxx + · · · is a restricted Euler operator, and Γ is proportional to

utx − g(u) and differential consequences. Thus we obtain the multiplier

Λ = Eux(Φt). (2.131)

Conversely, given a multiplier Λ, we can invert the relation (2.131) using Eq. (2.130) to


obtain the conserved density

Φt =

∫ 1

0

dλ(ux − ux)Λ[λu + (1− λ)u] (2.132)

where Λ[u] = Λ(x, u, ∂xu, . . . , ∂pxu), and u = u(x) is any function chosen so that the

expressions Λ[u] and g(u) are non-singular. In particular, if Λ[0] and g(0) are non-

singular, then we can choose u = 0, which simplifies the integral (2.132).

Thus, from Eqs. (2.131) and (2.132), we see that all nontrivial Klein-Gordon conservation

laws (2.121) of order q are determined by multipliers (2.122) of order p = 2q − 1 which

are obtained as solutions of the augmented system of symmetry determining equations

(2.126) to (2.129).

Through the determining system (2.126) to (2.129), we now derive a classification

of nonlinear Klein-Gordon interactions g(u) that admit higher-order conservation laws

(2.121) starting at order q = 2. The classification results are summarized by three theorems.

Theorem 2.3.1 The nonlinear Klein-Gordon equation (2.105) admits a conservation law

(2.121) of order q = 2 iff the nonlinear interaction is given by one of

g(u) = eu, g(u) = sin u, g(u) = eu ± e−u (2.133)

to within scaling of t, x, u, and translation of u.

(If we allow u to undergo complex-valued scalings and translations then the Klein-Gordon

equations for g(u) = sin u and g(u) = eu ± e−u are equivalent.)

The Klein-Gordon equations arising from Theorem 2.3.1 are the Liouville equation

(2.119), sine-Gordon equation (2.116) and related sinh-Gordon equation (2.117). (The

Tzetzeica equation (2.118) is absent in this classification because its first admitted higher-

order conservation law (2.121) is of order q = 3.) Since each of these Klein-Gordon

equations is known to admit an infinite sequence of higher order conservation laws

(2.121) for q > 2, this leads to an integrability classification.

Corollary 2.3.2 A nonlinear Klein-Gordon equation (2.105) is integrable in the sense of

having an infinite number of conservation laws (2.121) up to arbitrarily high orders q > 1 if

it admits one of order q = 2.

Theorem 2.3.3 The multipliers for the second-order conservation laws (2.121) admitted by

the Klein-Gordon equation (2.105) with nonlinear interactions (2.133) are given by

Λ = uxxx +1

2ux

3, (2.134)

Λ = uxxx − 1

2ux

3, (2.135)

for the sine-Gordon equation (2.116) and sinh-Gordon equation (2.117), respectively, and

Λ = (uxxx − uxuxx)fξ + fx + uxf = Dxf + uxf, (2.136)


depending on an arbitrary function

f = f(x, ξ), ξ = uxx − 1

2ux

2, (2.137)

for the Liouville equation (2.119).

From the construction formula (2.132) the corresponding conserved densities for the

multipliers (2.134) and (2.135), respectively, are given by

Φt =1

2uxxxux ± 1

8ux

4 = −1

2uxx

2 ± 1

8ux

4 + Dxθ, (2.138)

where we have used u = 0, since Λ[0] = 0 and g(0) = const. Similarly, the conserved

density for the multiplier (2.136) is given by

Φt = (ux − ux)Dx∫ 1

0

dλf(x, λ(uxx − uxx)− 1

2λ2(ux − ux)2)

+(ux − ux)2

∫ 1

0

dλ λf(x, λ(uxx − uxx)− 1

2λ2(ux − ux)2)

= −∫ uxx− 1

2 ux2

uxx− 12 ux

2

f(x, ξ)dξ + Dxθ, (2.139)

where now u = u(x) is any function chosen so that Λ[u] = Dxf(x, ξ) + uxf(x, ξ) and

g(u) = exp(u) are non-singular expressions.

We remark that the existence of the conservation law (2.139) involving an arbitrary func-

tion reflects the well-known classical integrability [9] (in the sense of explicit integration)

of the Liouville equation.

Proof of Theorem 2.3.1 From Eqs. (2.121) and (2.122), conservation laws with

Φt(x, u, ux, uxx) and Φx(x, u, ux, uxx) of order q = 2 correspond to multipliers

Λ(x, u, ux, uxx, uxxx). (2.140)

We start with the extra determining equations (2.127) to (2.129). These become

0 = 2Λu + D2xΛuxx − D3

xΛuxxx , (2.141)

0 = DxΛuxx − D2xΛuxxx , (2.142)

0 = Λuxx − DxΛuxxx . (2.143)

Note that Eq. (2.142) is a differential consequence of Eq. (2.143), while Eq. (2.141)

combined with a differential consequence of Eq. (2.142) reduces to

0 = Λu. (2.144)

Consider Eq. (2.143). Its highest-order term is Λuxxxuxxxuxxxx, and hence Λuxxxuxxx = 0. This

yields, using Eq. (2.144),

Λ = a(x, ux, uxx)uxxx + b(x, ux, uxx). (2.145)

Then the remaining terms in Eq. (2.143) give

buxx = ax + auxuxx. (2.146)


We now turn to the symmetry determining equation (2.126). This becomes, taking into

account Eqs. (2.140) and (2.144),

0 = DtΛx + uxxDtΛux + uxxxDtΛuxx + uxxxxDtΛuxxx+ΛuxDxg + ΛuxxD

2xg + ΛuxxxD

3xg − Λg′. (2.147)

We substitute Eq. (2.145) into Eq. (2.147) and separate it into highest derivative terms in

descending order. The terms of highest order involve uxxxx, which yield

0 = auxg + auxxuxg′. (2.148)

Since g depends only on u, while a does not depend on u, the terms in Eq. (2.148) can

balance only in two ways: either g and g′ are proportional, or g and g′ are linearly

independent with vanishing coefficients. This leads to two cases to consider.

Case (i): g, g′ linearly independent

In this case aux = auxx = 0, so a = a(x). Consequently from Eq. (2.146), we have buxx = ax,

and hence

b = a′(x)uxx + c(x, ux). (2.149)

To proceed, we return to the symmetry determining equation (2.147). The highest order

remaining terms now involve uxx, which lead to

0 = a′g′ + 3auxg′′ + cuxuxg. (2.150)

By taking a partial derivative with respect to ux we obtain 0 = 3ag′′ + cuxuxuxg. Since g

depends only on u, while a does not depend on u, this equation immediately yields

g′′ = σg, σ = const, (2.151)

cuxuxux = −3aσ, (2.152)

where σ� 0 in order for g(u) to be nonlinear. From Eq. (2.152) we obtain

c = −1

2σa(x)ux

3 + c2(x)ux2 + c1(x)ux + c0(x). (2.153)

Now we find that Eq. (2.150) reduces to 0 = a′g′+ 2c2g, which immediately separates into

the equations

a′ = c2 = 0. (2.154)

Finally, the terms remaining in the symmetry determining equation (2.147) simplify to

0 = c1xg − c0g′, and thus we have

c0 = c1x = 0. (2.155)

From Eq. (2.145), Eq. (2.149) and Eqs. (2.153) to (2.155), we obtain

Λ = a

(uxxx − 1

2σux

3

)+ c1ux, a = const, c1 = const. (2.156)

This multiplier is admitted for all nonlinear interactions g(u) satisfying Eq. (2.151) with


g(u) and g′(u) being linearly independent. The general solution for g(u) breaks into two

forms:

g(u) = α sin(√|σ|u+ β), if σ < 0; (2.157)

g(u) = α(e√σu+β ± e−√σu−β), if σ > 0; (2.158)

with arbitrary constants α, β, σ� 0. By a scaling and a translation√|σ|u + β → u, and

a scaling α√|σ|t → t, we see that the resulting Klein-Gordon equation (2.105) becomes,

respectively, the sine-Gordon equation (2.116) or the sinh-Gordon equation (2.117). This

completes the classification in case (i).

Case (ii): g, g′ proportional

In this case, we have

g′ = σg, σ = const (2.159)

where σ� 0 for g(u) to be nonlinear. Now Eq. (2.148) becomes

0 = aux + σauxxux. (2.160)

This is a linear first-order PDE for a in ux, uxx. The solution is

a = a(x, ξ), ξ = uxx − 1

2σux

2. (2.161)

Then from Eq. (2.146) we obtain

b = ax(x, ξ) + σux(a(x, ξ)− a(x, ξ)uxx) + c(x, ux), a =

∫adξ. (2.162)

To proceed, we return to the symmetry determining equation (2.147). We find that the

highest order terms involving uxxx2 and uxxx vanish as a consequence of Eq. (2.160). Then

we find that the next highest order terms which involve uxx in Eq. (2.147) yield cuxux = 0.

Hence, we obtain

c = c1(x)ux + c0(x). (2.163)

Finally, the remaining terms in Eq. (2.147) reduce to

0 = c′1 − σc0. (2.164)

Thus, from Eq. (2.145), Eqs. (2.161) and (2.162), Eqs. (2.163) and (2.164), we obtain

Λ = aξuxxx + σux(a− uxxaξ) + ax + c1ux +1

σc′1, ξ = uxx − 1

2σux

2, (2.165)

with arbitrary functions c1(x), a(x, ξ). This multiplier is admitted for all nonlinear interac-

tions g(u) satisfying Eq. (2.159). The general solution of this equation is

g(u) = eσu+β (2.166)

with arbitrary constants β, σ� 0. By a scaling and a translation σu+β → u, and a scaling

σt → t, we see that the resulting Klein-Gordon equation (2.105) becomes the Liouville

equation. This completes the classification in Case (ii). q


Acknowledgements

The authors are supported in part by the Natural Sciences and Engineering Research

Council of Canada. We gratefully thank the referees for useful comments which have

improved this paper.

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